A new (and optimal) result for the boundedness of a solution of a quasilinear chemotaxis–haptotaxis model (with a logistic source)

Abstract

This article deals with an initial-boundary value problem for the coupled chemotaxis–haptotaxis system with nonlinear diffusion(0.1){ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,τvt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂RN(N≥1), where τ∈{0,1} and χ, ξ, and μ are given nonnegative parameters. The diffusivity D(u) is assumed to satisfyD(u)≥CD(u+1)m−1for allu≥0andCD>0. In the present work it is shown that ifm≥2−2Nλwith0<μ<κ0,m>2−2Nλwithμ≥κ0, orm>2−2Nandμ=0 orm=2−2NandCD>CGN(1+‖u0‖L1(Ω))34(2−2N)2κ0, then for all reasonably regular initial data, a corresponding initial-boundary value problem for (0.1) possesses a unique global classical solution that is uniformly bounded in Ω×(0,∞), whereλ=κ0(κ0−μ)+ andκ0={maxs≥1⁡λ01s+1(χ+ξ‖w0‖L∞(Ω))ifτ=1,χifτ=0. Here CGN and λ0 are constants that correspond to the Gagliardo–Nirenberg inequality and the maximal Sobolev regularity, respectively. With use of new Lp-estimate techniques to obtain the a priori estimate of a solution from L1(Ω)→Lλ−ε(Ω)→Lλ(Ω)→Lλ+ε(Ω)→Lp(Ω)(for allp>1), these results significantly improve or extend previous results obtained by several authors.